Non-Linear Functions
5. Modeling with Functions
Learning outcomes
- I can choose an appropriate function type.
- I can justify my choice of model.
- I can interpret model predictions.
- I can evaluate limitations of models.
- I can solve practical problems using function models.
Modeling with Functions
A mathematical model uses mathematics to represent a real-world situation. Functions are especially useful because they describe how one quantity changes in relation to another.
For example, functions can model:
- The cost of a taxi journey
- The height of a ball after it is thrown
- The growth of a population
- The value of an investment
- The temperature of an object over time
- The amount of fuel used during a journey
The goal is not simply to find an equation. We need to choose a function that reasonably represents the pattern in the real situation.
What Is a Function Model?
A function model is a function used to represent the relationship between two quantities.
One quantity is usually the independent variable.
The other is the dependent variable because its value depends on the independent variable.
Suppose a bicycle rental company charges $8 plus $4 per hour.
Let:
t = rental time in hours
C = total cost in dollars
A model is:
C = 4t + 8
The model allows us to calculate the cost for different rental times.
For 3 hours:
C = 4(3) + 8
C = 20
The predicted cost is $20.
Choosing an Appropriate Function Type
Different situations require different types of functions.
Three common models are:
- Linear
- Quadratic
- Exponential
The pattern in the data and the context of the situation help determine which model is appropriate.
Linear Models
A linear model is appropriate when a quantity changes at approximately a constant rate.
A common form is:
y = mx + b
where:
m = constant rate of change
b = initial value
Consider a water tank containing 100 L of water. Water enters at a constant rate of 20 L per minute.
A model is:
V = 20t + 100
where:
V = volume in litres
t = time in minutes
The model is linear because the volume increases by the same amount every minute.
Clues That Suggest a Linear Model
Look for:
- Constant rate of change
- Constant first differences
- Approximately straight-line data
- The same amount being added or removed repeatedly
Examples include constant speed, hourly wages, constant filling rates and fixed cost per item.
Quadratic Models
A quadratic model is useful when the rate of change itself changes in a regular way.
A common form is:
y = ax² + bx + c
Its graph is a parabola.
One common example is projectile motion.
Imagine a ball thrown upward from a height of 2 m.
Its height might be modeled approximately by:
h = −4.9t² + 15t + 2
where:
h = height in metres
t = time in seconds
The ball initially rises, reaches a maximum height and then falls.
Clues That Suggest a Quadratic Model
Look for:
- A parabolic graph
- A quantity that increases and then decreases
- A quantity that decreases and then increases
- Constant second differences for equally spaced inputs
- Situations involving certain types of area or projectile motion
Exponential Models
An exponential model is appropriate when a quantity changes by approximately the same factor or percentage over equal intervals.
A common form is:
y = abˣ
where:
a = initial value
b = growth or decay factor
Suppose a bacterial population begins with 500 bacteria and doubles every hour.
A model is:
P = 500(2ᵗ)
After 1 hour:
P = 500(2¹) = 1000
After 2 hours:
P = 500(2²) = 2000
After 3 hours:
P = 500(2³) = 4000
The population increases by different amounts each hour, but it always multiplies by the same factor.
Clues That Suggest an Exponential Model
Look for:
- Constant percentage change
- Constant multiplying factor
- Constant ratio for equally spaced inputs
- Rapidly increasing growth
- Exponential decay toward zero
Examples include simplified population growth, compound interest, radioactive decay and depreciation.
Comparing the Models
The choice of model depends on how the quantity changes.
| Pattern | Likely Model |
|---|---|
| Constant amount added | Linear |
| Constant amount removed | Linear |
| Constant first difference | Linear |
| Constant second difference | Quadratic |
| Parabolic behaviour | Quadratic |
| Constant multiplying factor | Exponential |
| Constant percentage change | Exponential |
A useful question is:
What stays constant?
If the difference stays constant → think linear.
If the second difference stays constant → think quadratic.
If the ratio or percentage change stays constant → think exponential.
Choosing a Model from Data
Suppose a company records the following sales:
| Month | Sales |
|---|---|
| 1 | 100 |
| 2 | 120 |
| 3 | 140 |
| 4 | 160 |
| 5 | 180 |
The differences are:
+20, +20, +20, +20
Therefore, a linear model is appropriate.
Now consider:
| Month | Sales |
|---|---|
| 1 | 100 |
| 2 | 120 |
| 3 | 144 |
| 4 | 172.8 |
| 5 | 207.36 |
Each value is approximately:
previous value × 1.2
This represents growth of 20% per month.
Therefore, an exponential model is appropriate.
Justifying the Choice of Model
Choosing a model is only part of the process. We should also be able to explain why the model is appropriate.
A strong justification identifies the mathematical pattern.
For example:
A linear model is appropriate because the distance increases by approximately the same amount during each equal time interval.
Or:
An exponential model is appropriate because the population increases by approximately the same percentage each year.
A weak justification would simply say:
It looks exponential.
Instead, identify the evidence that supports the choice.
Using Models to Make Predictions
Once a model has been created, it can be used to estimate unknown values.
Suppose the number of visitors to a website is modeled by:
V = 500 + 75t
where t represents the number of months.
Predict the number of visitors after 8 months.
Substitute:
t = 8
V = 500 + 75(8)
V = 500 + 600
V = 1100
The model predicts approximately 1100 visitors after 8 months.
The word approximately is important because a mathematical model may not perfectly predict what actually happens.
Interpreting Predictions
A calculated answer should always be interpreted in the context of the problem.
Suppose a population model gives:
P(10) = 24,580
Do not simply report:
24,580
Instead explain:
The model predicts that the population will be approximately 24,580 after 10 years.
A complete interpretation should identify:
- What the number represents
- The appropriate units
- The input value used
- Whether the answer is exact or a prediction
Interpolation and Extrapolation
Predictions can be made within or outside the range of observed data.
Interpolation
Interpolation means estimating a value within the range of known data.
Suppose measurements were collected between 0 and 10 minutes.
Using the model to estimate a value at 6 minutes is interpolation.
Interpolation is often relatively reliable because the prediction lies within the region where data were collected.
Extrapolation
Extrapolation means predicting beyond the range of observed data.
Using the same data to predict what happens after 100 minutes would be extrapolation.
Extrapolation is generally more uncertain because the relationship may change outside the observed range.
Models Have Limitations
Mathematical models simplify reality.
A model may work very well under certain conditions but poorly under others.
For example, suppose a town has a population of 10,000 and grows by 5% every year.
An exponential model might be:
P = 10000(1.05)ᵗ
The model assumes that the population continues growing at exactly 5% indefinitely.
In reality, population growth can be affected by:
- Housing availability
- Jobs
- Migration
- Birth rates
- Death rates
- Government policies
- Economic conditions
- Natural disasters
Therefore, the model may be useful for short-term predictions but unreliable far into the future.
Models May Produce Impossible Predictions
One way to evaluate a model is to ask:
Does the prediction make sense in the real world?
Suppose a water tank contains 100 L and drains at 5 L per minute.
A linear model might be:
V = 100 − 5t
After 10 minutes:
V = 50 L
This makes sense.
After 20 minutes:
V = 0 L
This also makes sense.
But after 30 minutes:
V = −50 L
A tank cannot contain −50 L of water.
The mathematics continues, but the real-world model should stop when the tank reaches zero.
Therefore, the useful domain of the model is:
0 ≤ t ≤ 20
This illustrates why the domain of a mathematical model is often restricted by the real situation.
Assumptions in Models
Models are usually based on assumptions.
An assumption is something we accept as true in order to simplify a situation.
For example, a constant-speed travel model may assume:
- The vehicle maintains the same speed
- There are no stops
- Traffic conditions do not change
- The route does not change
These assumptions make the mathematics easier.
However, they also create limitations.
If the assumptions become unrealistic, the model becomes less reliable.
Worked Example: Taxi Fare
A taxi company charges a $5 starting fee plus $3 for every kilometre travelled.
Choose a Model
The fare increases by a constant amount for every kilometre.
Therefore, a linear model is appropriate.
Let:
d = distance travelled
C = total cost
The model is:
C = 3d + 5
Make a Prediction
Find the cost of travelling 12 km.
C = 3(12) + 5
C = 36 + 5
C = $41
Interpret the Result
The model predicts that a 12 km taxi journey will cost $41.
Evaluate the Model
The model assumes that:
- The rate remains $3/km
- There are no additional fees
- There are no waiting-time charges
- The pricing system does not change
If these assumptions are incorrect, the actual fare may differ.
Worked Example: Depreciation
A computer is purchased for $2,000 and loses 20% of its value each year.
Because the value changes by the same percentage, an exponential model is appropriate.
The yearly multiplier is:
1 − 0.20 = 0.80
Therefore:
V = 2000(0.80)ᵗ
After 3 years:
V = 2000(0.80)³
V = 2000(0.512)
V = 1024
The model predicts a value of approximately $1,024 after three years.
Limitation
The object will probably not continue losing exactly 20% of its value forever.
Eventually, other factors such as condition, rarity, demand or minimum resale value may become more important.
Worked Example: Projectile Motion
Suppose the height of a ball is modeled by:
h = −5t² + 20t + 1
where:
h = height in metres
t = time in seconds
Find its height after 2 seconds.
h = −5(2²) + 20(2) + 1
h = −20 + 40 + 1
h = 21 m
The model predicts that the ball is approximately 21 metres above the reference level after 2 seconds.
This model is quadratic because the variable t is squared.
Limitation
The model may assume simplified gravitational motion and may ignore effects such as air resistance and wind.
The Modeling Process
Function modeling can be viewed as a cycle.
A useful process is:
Understand the situation → Identify variables → Examine the pattern → Choose a function → Build the model → Make predictions → Interpret results → Evaluate the model
If the model does not fit the data or produces unrealistic predictions, it may need to be changed.
Evaluating Whether a Model Is Good
A useful model should:
- Fit the available data reasonably well
- Produce realistic predictions
- Match the expected mathematical pattern
- Have reasonable assumptions
- Be useful within an appropriate domain
Ask questions such as:
Does the model fit the observed data?
Does the prediction make sense?
Are the assumptions reasonable?
How far beyond the data are we predicting?
Could another function provide a better model?
A model does not need to be perfect to be useful.
Common Mistakes
Choosing a function only because the graph looks similar
Also examine differences, ratios, rates of change and the context.
Assuming models make exact predictions
Most real-world models provide estimates rather than guaranteed outcomes.
Ignoring units
A prediction such as "35" is incomplete if we do not know whether it represents dollars, kilometres, people or seconds.
Extrapolating too far
A model that fits current data may become unreliable far beyond the observed range.
Ignoring unrealistic results
A mathematically correct calculation can still produce a physically impossible prediction.
Forgetting assumptions
Every model simplifies reality. Understanding its assumptions helps us judge when the model is useful.
Key Terms
Mathematical Model: A mathematical representation of a real-world situation.
Function Model: A function used to represent the relationship between quantities.
Independent Variable: The input variable used to determine another quantity.
Dependent Variable: The output variable whose value depends on the input.
Linear Model: A model with a constant rate of change.
Quadratic Model: A model represented by a quadratic function, often producing a parabolic graph.
Exponential Model: A model involving repeated proportional or percentage change.
Prediction: A value estimated using a mathematical model.
Interpolation: Estimating a value within the range of observed data.
Extrapolation: Predicting a value beyond the range of observed data.
Assumption: A condition accepted as true when constructing a model.
Limitation: A condition or weakness that restricts how accurately or widely a model can be used.
Domain: The set of input values for which a model is meaningful.
Rate of Change: The amount an output changes relative to a change in input.
Check Your Understanding
1. What is a mathematical model?
2. A worker earns $20 per hour. Which function type would be appropriate for modeling earnings versus time? Explain your choice.
3. A population increases by 4% each year. Which type of function would be appropriate? Why?
4. A ball rises, reaches a maximum height and falls. Which type of function could model its height versus time?
5. A tank contains 300 L and loses water at 10 L per minute. Write a function that models the amount of water remaining.
6. Use your model from Question 5 to predict the amount remaining after 12 minutes.
7. Why would the model in Question 5 eventually stop being useful?
8. Explain the difference between interpolation and extrapolation.
9. Why are predictions made far beyond the available data generally less reliable?
10. A laptop worth $1,500 loses 10% of its value each year. Write an exponential model for its value after t years.
11. Use the model to predict the laptop's value after 2 years.
12. Give one limitation of using the same depreciation model for the next 30 years.
Key Takeaways
- A mathematical model uses mathematics to represent a real-world relationship.
- Linear models are appropriate when quantities change by approximately a constant amount.
- Quadratic models are useful for relationships involving parabolic behaviour or steadily changing rates.
- Exponential models are appropriate when quantities change by approximately a constant factor or percentage.
- A model should be chosen using mathematical evidence and the context of the situation.
- A good justification explains why the pattern matches the chosen function type.
- Models can be used to make predictions and solve practical problems.
- Predictions should always be interpreted using the correct context and units.
- Interpolation makes predictions within the observed data range.
- Extrapolation makes predictions beyond the observed data range and is usually less reliable.
- Every mathematical model involves assumptions and therefore has limitations.
- A mathematically correct prediction may still be unrealistic in the real world.
- The meaningful domain of a model may be smaller than the mathematical domain of its equation.
- Good mathematical modeling involves choosing, using, interpreting and then evaluating the model—not simply finding an equation.